Alphabeta MathDerived Hall Algebras from dg-Enhanced Categories
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1 · Prerequisites
Abstract Algebra
Algebraic Topology
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Differential Graded Algebras and Homological Foundations
- Lefschetz Splitting In Bounded Triangulated Categories
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Triangulated Categories
- Yoneda Extensions and Homological Dimension
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
2 · Logical flowchart
3 · Items
Definition
Locally finite homotopy types and rational cardinality
Lemma
Homotopy-cardinality pushforward satisfies base change
Lemma
Homotopy-fiber weights reduce to componentwise ratios
Definition
Locally finite dg enhancements for derived Hall theory
Lemma
Homotopy-cardinality pushforward is functorial
Definition
Toën derived Hall algebra by homotopy-cofiber convolution
Lemma
Perfect dg modules have finite total Ext
Lemma
Derived Hall moduli types are locally finite and the input span is proper
Lemma
Mapping-space homotopy groups give the derived Hall stabilizer weight
Lemma
The two derived Hall correspondence squares are homotopy cartesian
Theorem
Toën derived Hall numbers have the higher-Ext orbit formula
Theorem
Toën derived Hall multiplication is associative and unital
Theorem
The opposite untwisted Hall algebra of a hereditary heart embeds in the derived Hall algebra
Remark
Toën, Kapranov, and unweighted triangle counts are distinct constructions
Sources
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §2
- Olivier Schiffmann, Lectures on Hall algebras, arXiv:math/0611617v2, §5.6
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §2, Lemma 2.6
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §2, Lemma 2.3
- Allen Hatcher, Algebraic Topology, §4.2, Theorem 4.41
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §§3–4
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §2, Corollary 2.4
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §3
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §3, Lemma 3.2
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §5, Proposition 5.1
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §4
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §4, Theorem 4.1
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §6, Proposition 6.1
- Bertrand Toën, Derived Hall algebras, arXiv:math/0501343v4, §§1 and 5–7